MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
1996 Korea National Olympiad
1996 Korea National Olympiad
Part of
Korea National Olympiad
Subcontests
(8)
8
1
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Geometry!
Let
△
A
B
C
\triangle ABC
△
A
BC
be the acute triangle such that
A
B
≠
A
C
.
AB\ne AC.
A
B
=
A
C
.
Let
V
V
V
be the intersection of
B
C
BC
BC
and angle bisector of
∠
A
.
\angle A.
∠
A
.
Let
D
D
D
be the foot of altitude from
A
A
A
to
B
C
.
BC.
BC
.
Let
E
,
F
E,F
E
,
F
be the intersection of circumcircle of
△
A
V
D
\triangle AVD
△
A
V
D
and
C
A
,
A
B
CA,AB
C
A
,
A
B
respectively. Prove that the lines
A
D
,
B
E
,
C
F
AD, BE,CF
A
D
,
BE
,
CF
is concurrent.
7
1
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This is kinda weird problem...
Let
A
n
A_n
A
n
be the set of real numbers such that each element of
A
n
A_n
A
n
can be expressed as
1
+
a
1
2
+
a
2
(
2
)
2
+
⋯
+
a
n
(
n
)
n
1+\frac{a_1}{\sqrt{2}}+\frac{a_2}{(\sqrt{2})^2}+\cdots +\frac{a_n}{(\sqrt{n})^n}
1
+
2
a
1
+
(
2
)
2
a
2
+
⋯
+
(
n
)
n
a
n
for given
n
.
n.
n
.
Find both
∣
A
n
∣
|A_n|
∣
A
n
∣
and sum of the products of two distinct elements of
A
n
A_n
A
n
where each
a
i
a_i
a
i
is either
1
1
1
or
−
1.
-1.
−
1.
6
1
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min of k?
Find the minimum value of
k
k
k
such that there exists two sequence
a
i
,
b
i
{a_i},{b_i}
a
i
,
b
i
for
i
=
1
,
2
,
⋯
,
k
i=1,2,\cdots ,k
i
=
1
,
2
,
⋯
,
k
that satisfies the following conditions.(i) For all
i
=
1
,
2
,
⋯
,
k
,
i=1,2,\cdots ,k,
i
=
1
,
2
,
⋯
,
k
,
a
i
,
b
i
a_i,b_i
a
i
,
b
i
is the element of
S
=
{
199
6
n
∣
n
=
0
,
1
,
2
,
⋯
}
.
S=\{1996^n|n=0,1,2,\cdots\}.
S
=
{
199
6
n
∣
n
=
0
,
1
,
2
,
⋯
}
.
(ii) For all
i
=
1
,
2
,
⋯
,
k
,
a
i
≠
b
i
.
i=1,2,\cdots, k, a_i\ne b_i.
i
=
1
,
2
,
⋯
,
k
,
a
i
=
b
i
.
(iii) For all
i
=
1
,
2
,
⋯
,
k
,
a
i
≤
a
i
+
1
i=1,2,\cdots, k, a_i\le a_{i+1}
i
=
1
,
2
,
⋯
,
k
,
a
i
≤
a
i
+
1
and
b
i
≤
b
i
+
1
.
b_i\le b_{i+1}.
b
i
≤
b
i
+
1
.
(iv)
∑
i
=
1
k
a
i
=
∑
i
=
1
k
b
i
.
\sum_{i=1}^{k} a_i=\sum_{i=1}^{k} b_i.
∑
i
=
1
k
a
i
=
∑
i
=
1
k
b
i
.
5
1
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Famous ID from Korea
Find all integer solution triple
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
such that
x
2
+
y
2
+
z
2
−
2
x
y
z
=
0.
x^2+y^2+z^2-2xyz=0.
x
2
+
y
2
+
z
2
−
2
x
yz
=
0.
4
1
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Circle inside the angle
Circle
C
C
C
(the center is
C
C
C
.) is inside the
∠
X
O
Y
\angle XOY
∠
XO
Y
and it is tangent to the two sides of the angle. Let
C
1
C_1
C
1
be the circle that passes through the center of
C
C
C
and tangent to two sides of angle and let
A
A
A
be one of the endpoint of diameter of
C
1
C_1
C
1
that passes through
C
C
C
and
B
B
B
be the intersection of this diameter and circle
C
.
C.
C
.
Prove that the cirlce that
A
A
A
is the center and
A
B
AB
A
B
is the radius is also tangent to the two sides of
∠
X
O
Y
.
\angle XOY.
∠
XO
Y
.
3
1
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floor + sum
Let
a
=
⌊
n
⌋
a=\lfloor \sqrt{n} \rfloor
a
=
⌊
n
⌋
for given positive integer
n
.
n.
n
.
Express the summation
∑
k
=
1
n
⌊
k
⌋
\sum_{k=1}^{n}\lfloor \sqrt{k} \rfloor
∑
k
=
1
n
⌊
k
⌋
in terms of
n
n
n
and
a
.
a.
a
.
2
1
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FE integers
Let the
f
:
N
→
N
f:\mathbb{N}\rightarrow\mathbb{N}
f
:
N
→
N
be the function such that (i) For all positive integers
n
,
n,
n
,
f
(
n
+
f
(
n
)
)
=
f
(
n
)
f(n+f(n))=f(n)
f
(
n
+
f
(
n
))
=
f
(
n
)
(ii)
f
(
n
o
)
=
1
f(n_o)=1
f
(
n
o
)
=
1
for some
n
0
n_0
n
0
Prove that
f
(
n
)
≡
1.
f(n)\equiv 1.
f
(
n
)
≡
1.
1
1
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Well known PHP
If you draw
4
4
4
points on the unit circle, prove that you can always find two points where their distance between is less than
2
.
\sqrt{2}.
2
.